232 PHYSICS. 
angular distance. The light in these successive refractions and reflections 
becomes decomposed, and as each color has a different refractive power, it 
follows, that to see all the colors (violet to red) from the same drop it will 
be necessary to take up successive positions, so that the angle POz may 
range from 40° 17’ to 42° 2’. If, however, as is actually the case, we have 
a series of drops, one above the other, and as the angle made between the 
axis of the box and the line from the eye to the individual drops, amounts 
to from 40° 17' to 42° 2’, we shall, without changing our position, observe 
all these colors. The breadth of the arc is about 2°, the colors fading one 
into the other, and succeeding in the order of the spectrum, the red being 
uppermost. When the sun is veiled by mists or clouds the bow may 
appear simply as a light colored arc. The nearer to the horizon the sun 
stands the greater is the arc described ; when the sun is in the horizon the 
bow is a semicircle. An observer under favorable circumstances may, 
from an elevated position, see more than a semicircle, and completely 
circular bows can often be observed in the spray from fountains, waterfalls, 
&c. When the sun is 42° above the horizon no bow will be visible. 
_ Besides the primary bow just described, a secondary bow may almost 
always be observed cutside of the first, but still concentric with it, and with 
the colors in the reverse order. The red is here the inner color, and the 
violet the outer. The colored bands are broader but weaker than the 
corresponding ones of the primary bow. This secondary bow is produced by 
the colored rays from drops at such an elevation that they just reach the 
eye after undergoing two refractions and two reflections, while in the first 
case there are two refractions and but one reflection. The same drop 
cannot furnish at a given instant to the same position of the observer 
colors belonging to both bows, although it may in different stages of 
descent, or to different observers. The angular position of the second bow 
is between 50° 59’ and 54° 9’, with respect to the axis or the line from the 
sun through the eye to the spectator; there may be more than two bows 
resulting from rays that have undergone more than two reflections, but the 
light becomes finally so feeble as to render it impossible in ordinary cases 
to distinguish them. 
It is evident from the preceding principles that a rainbow may appear in 
the west in the morning, in the east in the evening, and in the north at 
noon ; never in the south. Also, that every observer sees his own rainbow, 
and that this changes for him with every successive instant. A drop of 
water can only play its part to a fixed observer who sees two bows during 
two periods of time, the first in passing the limits of the upper bow, the 
second in passing across the second. 
Sometimes other colors are added to the violet of the primary bow, as, 
for instance, a second green and second violet; red also is sometimes 
perceived. This same repetition of colors may also be observed occa- 
sionally on secondary bows. The cause of these irregular colors is not 
yet satisfactorily explained. 
Some different rainbow formations are exhibited in pl. 26, figs. 10 and 11. 
Should the image of the sun be reflected from the surface of extended calm 
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